On boundedness of zeros of the independence polynomial of tori

We study boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. We prove that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori. Here balanced means that the size of the torus is at most exponential in the shortest side length, while highly unbalanced means that the longest side length of the torus is super exponential in the product over the other side lengths cubed. We discuss implications of our results to the existence of efficient algorithms for approximating the independence polynomial on tori. This project was partially inspired by the relationship between zeros of partition functions and holomorphic dynamics, a relationship that in the last two decades played a prominent role in the field. Besides presenting new results, we survey this relationship and its recent consequences.
Comment: 51 pages, 8 figures. Section 1.3 has been expanded quite a bit and has been made into a new Section 2. Section 1.5 and Section 1.4 have swapped places. Some other small changes including added references